Self $C_2$-equivalence of two-component links and invariants of link maps

We use Kirk’s invariant of link maps $S^2\bigsqcup S^2\to S^4$ and its variations due to Koschorke and Kirk-Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi-Ohyama classification of two-component links in $S^3$ up to $\mathrm{\Delta }$-link homotopy. We also prove its version for string links, which is due (in a slightly different form) to Fleming-Yasuhara. The proofs do not use Clasper Theory.